PEWAUKEE, Wis. — Michelle Roelandts had a question for her congressman: If the Affordable Care Act and its premium subsidies were repealed, what would happen when her daughter turns 26 this year and needs to get her own health insurance while attending law school?

Representative Jim Sensenbrenner, a durable Wisconsin Republican who has served in the House since 1979, had little to offer in response. “If I could give you an answer today, I would, but I can’t,” Mr. Sensenbrenner said at a town-hall-style meeting on Saturday, where about 70 people packed a room at the Pewaukee Public Library.

Ms. Roelandts’s question and others like it are being asked with increasing anger and urgency across the country, and Republicans have found themselves on the defensive — for all their fury aimed at repealing the law, so far they have not agreed on an alternative.

Soon after, a man yelled to Mr. Sensenbrenner: “How many times did you vote to repeal without knowing what the replacement would be? How many times? Dozens!”

The congressman, who prides himself on his prolific schedule of town-hall-style meetings, banged his gavel and insisted that his rules for civility be obeyed.

While Mr. Sensenbrenner did not face the kind of anger that some of his peers did in recent days, he must answer the same question: Is this resistance a sign of a sustainable organic movement, or one that will soon flame out? And like his colleagues, he is also coming to grips with how much he will be saddled with the combative comments made by President Trump.

Mr. Sensenbrenner, in an interview, attributed the turnout at his gatherings to “organized opposition by people who were on the losing side of the election.”

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Mr. Sensenbrenner attributed the attendance at his gatherings to “organized opposition by people who were on the losing side of the election.”CreditLauren Justice for The New York Times

Facing restive audiences in public meetings is not new, but in the age of social media, an ugly scene in one congressional district can quickly attract widespread attention.

The questions from voters on display this weekend at a series of town-hall-style meetings in Wisconsin’s Fifth Congressional District, many of which were focused on the future of the health care law, underscored the quandary many lawmakers are facing even in solidly Republican districts.

The imminent problem: Constituents want answers, and without any consensus on how to go about replacing the law, Republicans have little to say.

“It’s kind of like, you know, getting a 30,000-piece jigsaw puzzle for Christmas,” Mr. Sensenbrenner said, “and, you know, cleaning off the dining room table and seeing how long it takes to put all the 30,000 pieces together in the right place. It’s not going to be easy.”

Mr. Sensenbrenner won re-election last year by 37 percentage points. His right-leaning district, which includes suburbs around Milwaukee, voted decisively for Mr. Trump over Hillary Clinton.

At three town-hall-style meetings over the weekend, Mr. Sensenbrenner sat at the front of the room to take questions from people who submitted slips of paper listing their name and address. When he called on people, he read their names and where they live — a practice that makes people “less likely to make fools of themselves,” he said in the interview.

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Citizens flooded in to register before one of Mr. Sensenbrenner’s gatherings at the Pewaukee Public Library on Saturday.CreditLauren Justice for The New York Times

At the meetings, he faced crowds that were adversarial but generally civil, and he fielded questions on a range of issues. At moments when the gatherings grew a bit unruly, he did not hesitate to bang his gavel. Like a frustrated grade-school teacher, he offered some unsolicited advice about comportment.

“This is not a session on who can cheer or boo the loudest,” he said as he began the Pewaukee meeting on Saturday, urging people to “be respectful of opinions that you do not share.”

The tough questioning of Republican lawmakers has been driven partly by concerns over health care, but also by outrage over Mr. Trump’s presidency.

That was true in Wisconsin, too. Mr. Sensenbrenner, who has long worked on immigration issues in Congress, said the executive order on immigration was “completely messed up” and a “train wreck.” And he suggested he would be of little assistance in reining in Mr. Trump.

“Do you think I’m able to control anybody else’s mouth, from the president on down?” he asked.

Repeatedly, the questions Mr. Sensenbrenner faced over the weekend showed the challenge that lawmakers have in explaining the effects of repealing the Affordable Care Act, especially now, when Republicans have yet to coalesce around a replacement plan.

Pressed by one questioner to oppose a replacement for the health care law if that replacement would raise costs for sick people, he explained that “there are winners and losers” when bills are passed.

A woman told him she learned she had skin cancer in 2005, and she asked about coverage for people with pre-existing conditions. Another woman told him that her 97-year-old mother was in a nursing home, and she wondered whether changing Medicaid to give each state a fixed amount of money, called a block grant, could cause her mother to be “put out on the street.”

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People who could not get in watched Mr. Sensenbrenner’s town-hall-style meeting in the hallway of the library in Pewaukee. Republican representatives have faced protesters at similar gatherings recently.CreditLauren Justice for The New York Times

Leigh Levas, 35, a medical technologist, told him that her 9-year-old daughter had juvenile rheumatoid arthritis.

“I’ve been sending him postcards with her photo, because I think he needs to see the people that it affects,” Ms. Levas said after the Pewaukee meeting.

Mr. Sensenbrenner offered a reassurance that some popular aspects of the health care law would remain: insurers would not be able to deny coverage because of pre-existing conditions, young adults could stay on their parents’ health plan until they are 26, and lifetime limits on coverage would not be allowed. And he acknowledged the stakes of the repeal effort.

“From a political standpoint, we Republicans know that we will own whatever the replacement will be, just as Obama and the Democrats own the A.C.A.,” he told Ms. Roelandts, who asked about health coverage for her daughter. “We got to get it right, and we got to get it right the first time.”

Ms. Roelandts, an accountant, said later that she was not happy with his answer. “I kind of interpret it as they don’t really know what they’re going to do yet,” she said, adding that she was alarmed by the comparison to a 30,000-piece jigsaw puzzle.

“Don’t talk about repealing something until you have valid ideas on the table for replacing it,” she said. “I mean, it’s causing me to literally lose sleep at night.”

Still, Mr. Sensenbrenner was blunt and unapologetic about the Republican push for dismantling President Barack Obama’s signature domestic achievement.

“I won by 146,000 votes,” he said in the interview. “I represent the majority. Now, they’re a vocal minority.”